Kato expansion in quantum canonical perturbation theory

Abstract

This work establishes a connection between canonical perturbation series in quantum mechanics and a Kato expansion for the resolvent of the Liouville superoperator.Our approach leads to an explicit expression for a generator of a block-diagonalizing Dyson's ordered exponential in arbitrary perturbation order. Unitary intertwining of perturbed and unperturbed averaging superprojectors allows for a description of ambiguities in the generator and block-diagonalized Hamiltonian.We compare the efficiency of the corresponding computational algorithm with the efficiencies of the Van Vleck and Magnus methods for high perturbative orders.

Superoperator Master Equations and Effective Dynamics

“…Thus, after the Bogolubov-van Hove scaling in the limit λ → 0, the generator becomes a generator of fully unitary dynamics without any dissipator-like terms, which is not obvious from the abstract form (68). Let us also remark that there is a connection of Formula (69) with the algebraic perturbation theory [30,32,52]. The algebraic perturbation theory for given L 0 and L I finds perturbatively such U (λ) and L eff,sec (λ) that…”

“…Then, taking the integral with respect to s in ( 15 ), similarly to [ 40 ], (Corollary 1), we have Here, is a pseudoinverse such that where is a hyperprojector to the kernel of the map , i.e., it is zero on the kernel and the inverse in the usual sense on the orthogonal complement to the kernel. Such a pseudoinverse can be defined by the explicit formula [ 52 ], (Equation ( 2 )) Thus, within the accuracy of the second order of perturbation theory, the projected propagator can be defined as a solution of the Cauchy problem …”

“…Thus, after the Bogolubov–van Hove scaling in the limit , the generator becomes a generator of fully unitary dynamics without any dissipator-like terms, which is not obvious from the abstract form ( 68 ). Let us also remark that there is a connection of Formula ( 69 ) with the algebraic perturbation theory [ 30 , 32 , 52 ]. The algebraic perturbation theory for given and finds perturbatively such and that where and is a unitary transformation and contains only the “secular” terms, which can be formalized [ 52 ] as the condition Comparing with Formula (13) from [ 52 ] for the effective Hamiltonian, we obtain However, let us remark that after the Bogolubov–van Hove scaling, the unitary transformation , which can also be found in [ 52 ], (Section IV.B) seems to play no role.…”

“…Let us also remark that there is a connection of Formula ( 69 ) with the algebraic perturbation theory [ 30 , 32 , 52 ]. The algebraic perturbation theory for given and finds perturbatively such and that where and is a unitary transformation and contains only the “secular” terms, which can be formalized [ 52 ] as the condition Comparing with Formula (13) from [ 52 ] for the effective Hamiltonian, we obtain However, let us remark that after the Bogolubov–van Hove scaling, the unitary transformation , which can also be found in [ 52 ], (Section IV.B) seems to play no role. This is because in ( 69 ) has higher order corrections , so with the same accuracy, and the unitary transformation can be neglected.…”

“…where P ker[L 0 , • ] is a hyperprojector to the kernel of the map [L 0 , • ], i.e., it is zero on the kernel and the inverse in the usual sense on the orthogonal complement to the kernel. Such a pseudoinverse can be defined by the explicit formula [52], (Equation ( 2))…”